Centrality in Social Networks Kristina Lerman University of

Centrality in Social Networks Kristina Lerman University of Southern California CS 599: Social Media Analysis University of Southern California 1

Network analysis

4. 2

Three views of social network analysis 1. Freeman, L. 1979 “Centrality in Social Networks: Conceptual Clarification”, Social Networks 1, No. 3. 2. Bonacich, P. 1987 “Power and Centrality in Social Networks: a Family of Measures”, American Journal of Sociology 3. Franceschetti, M. 2011 “Page. Rank: standing on the shoulders of giants” Commun. ACM, Vol. 54, pp. 92 -101.

Power in social networks Certain positions within the network give nodes more power – Directly affect/influence others – Control the flow of information – Avoid control of others

Centrality in social networks • Centrality encodes the relationship between structure and power in groups Certain positions within the network give nodes more power or importance • How do we measure importance? – Who can directly affect/influence others? • Highest degree nodes are “in the thick of it” – Who controls information flow? • Nodes that fall on shortest paths between others can disrupt the flow of information between them – Who can quickly inform most others? • Nodes who are close to other nodes can quickly get information to them

Degree centrality • The number of others a node is connected to – Node with high degree has high potential communication activity node In-degree Outdegree Total degree 1 0 1 1 2 3 2 5 3 1 3 4 4 2 1 3 5 2 1 3 4 5 2 3 1 4 5 1 2 3

Mathematical representation and operations on graphs Adjacency matrix A 1 2 3 4 5 1 0 0 0 2 0 0 1 1 0 3 0 1 1 4 0 0 1 5 0 1 0 0 0 5 2 3 1 node In-degree Outdegree Total degree 1 0 1 1 2 3 2 5 3 1 3 4 4 2 1 3 5 2 1 3 Out-degree: row sum In-degree: column sum 4

Betweenness centrality • Number of shortest paths (geodesics) connecting all pairs of other nodes that pass through a given node – Node with highest betweenness can potentially control or distort communication 1 2 3 1 2 4 5 2 3 2 4 5 3 2 3 4 3 5 4 5 2 3 4 5 5 2 4 4 5 2 3 1 4 5 1 2 3

Closeness centrality • Node that is closest to other nodes can reach other nodes in shortest amount of time – Can best avoid being controlled by others • Closeness centrality is sum of geodesic distances from a node to all other nodes

Self-consistent measures of centrality • Katz (1953): Katz score – “not only on how many others a person is connected to, but who he is connected to” – One’s status is determined by the status of the people s/he is connected to • Bonacich (1972): Eigenvector centrality – Node’s centrality is the sum of the centralities of its connections – e is the eigenvector of A, and l its associated eigenvalue (largest eigenvalue)

Alpha-centrality (Bonacich, 1987) • Similar to eigenvector centrality, but the degree to which a node centrality contributes to the centralities of other nodes depends on parameter a. • Mathematical interpretation – ci(a) is the expected number of paths activated directly or indirectly by a node i

A closer look at Alpha-Centrality • Alpha-Centrality matrix 4 5 2 3 1 • 1 st term: number of paths of length 1 (edges) between i and j 0 1 0 0 0 1 1 0 0 1 0 0 0 • Contribution of this term to ci(a) is Sj. Aij

A closer look at Alpha-Centrality • Alpha-Centrality matrix 4 5 2 3 1 • 2 nd term: number of paths of length 2 between i and j 0 1 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 x = 0 0 1 1 0 0 1 2 0 1 1 0 1 0 0 0 1 1 0

A closer look at Alpha-Centrality • Alpha-Centrality matrix 4 5 2 3 1 • 3 rd term: number of paths of length 3 between i and j 0 0 1 1 0 0 0 0 1 0 1 2 0 0 1 1 0 0 2 1 1 1 0 1 0 1 1 0 2 1 2 2 0 1 0 0 0 0 1 1 0 0 1 0 1 2 x =

A closer look at Alpha-Centrality • Alpha-Centrality matrix • Number of paths of diverges as length of the path k grows • To keep the infinite sum finite, a < 1/l 1, where l 1 is the largest eigenvalue of A (also called radius of centrality) • Interpretation: Node’s centrality is the sum of paths of any length connecting it to other nodes, exponentially attenuated by length of the path, so that longer paths contribute less than shorter paths

Radius of centrality Parameter a sets the length scale of communication or interactions. • For a = 0, only local interactions (with neighbors) are considered – Only local structure is important – centrality is same as degree centrality

Radius of centrality Parameter a sets the length scale of communication or interactions. • As a grows, the length of interaction grows – Global structure becomes more important – Centrality depends on node’s position within a larger structure, e. g. , a community

Radius of centrality Parameter a sets the length scale of communication or interactions. • As a 1/l 1, length of interactions becomes infinite – Global structure is important – Centrality is same as eigenvector centrality

Normalized Alpha-Centrality [Ghosh & Lerman 2011] • Alpha-Centrality diverges for a > 1/l 1 • Solution: Normalized Alpha-Centrality – Holds for

Multi-scale analysis with Alpha-Centrality • Parameter a allows for multi-scale analysis of networks – Differentiate between local and global structures • Study how rankings change with a • Leaders: high influence on group members • Nodes with high centrality for small values of a • Bridges: mediate communication between groups • Nodes with low centrality for small values of a • But high centrality for large values of a • Peripherals: poorly connected to everyone • Nodes with low centrality for any value of a

Karate club network [Zachary, 1977] administrator instructor

Ranking karate club members Centrality scores of nodes vs. a

Florentine families in 15 th century Italy

Ranking Florentine families

Summary • Network position confers advantages or disadvantages to a node, but how you measure it depends on what you mean by advantage – Ability to directly reach many nodes degree centrality – Ability to control information betweenness centrality – Ability to avoid control closeness centrality • Self-consistent definitions of centrality – Node’s centrality depends on centrality of those it is connected to, directly or indirectly, but contribution of distant nodes is attenuated by how far they are • Attenuation parameter sets the length scale of interactions • Can probe structure at different scales by varying this parameter

Page. Rank: Standing on the Shoulders of Giants [Franceschet] Key insights • Analyzes the structure of the web of hyperlinks to determine importance score of web pages – A web page is important if it is pointed to by other important pages • An algorithm with deep mathematical roots – Random walks – Social network theory

Page. Rank and the Random Surfer • Starts at arbitrary page I H F L M G B C E D A

Page. Rank and the Random Surfer • Starts at arbitrary page • Bounces from page to page by following links randomly I H F L M G B C E D A

Page. Rank and the Random Surfer I H F L M G B C E D A Random Surfer • Starts at arbitrary page • Bounces from page to page by following links randomly • Page. Rank score of a web page is the relative number of time it is visited by the Random Surfer

Mathematics of Page. Rank • Page. Rank is a solution to a random walk on a graph Adjacency matrix of the graph A I H F L M G B C E D A A B C D E F G H I L M A 0 0 0 B 0 0 1 0 0 0 0 C 0 1 0 0 0 0 0 D 1 1 0 0 0 0 0 E 0 1 0 1 0 0 0 F 0 1 0 0 0 0 G 0 1 0 0 0 0 H 0 1 0 0 0 0 I 0 1 0 0 0 0 L 0 0 1 0 0 0 M 0 0 1 0 0 0

Mathematics of Page. Rank • Page. Rank is a solution to a random walk on a graph I (Diagonal) Out-degree matrix D H F L M G B C E D A A B C D E F G H I L M A 0 0 0 B 0 1 0 0 0 0 0 C 0 0 1 0 0 0 0 D 0 0 0 2 0 0 0 0 E 0 0 3 0 0 0 F 0 0 0 2 0 0 0 G 0 0 0 2 0 0 H 0 0 0 0 2 0 0 0 I 0 0 0 0 2 0 0 L 0 0 0 0 0 1 0 M 0 0 0 0 0 1

Mathematics of Page. Rank • Page. Rank is a solution to a random walk on a graph • hij is probability to go from node i to node j hij=1/di H=D-1 A I H F L M G B C E D A A B C D E F G H I L M A 0 0 0 B 0 0 1 0 0 0 0 C 0 1 0 0 0 0 0 D . 5 0 0 0 0 0 E 0 . 3 0 0 0 F 0 . 5 0 0 1 . 5 0 0 0 G 0 . 5 0 0 0 0 H 0 . 5 0 0 0 0 I 0 . 5 0 0 0 0 L 0 0 1 0 0 0 M 0 0 1 0 0 0

Mathematics of Page. Rank • Page. Rank of page j is defined recursively as I – pj=Sipihij – Or in matrix form p=p. H H • What contributes to Page. Rank score? L – Number of links page j receives • Cf B and D – Number of outgoing links of linking pages • Cf E’s effect on F and B’s effect on C – Page. Rank scores of linking pages • Cf E and B M G F B C D A E

… but there are problems • Random Surfer gets trapped by dangling nodes! (no outlinks) • Solution: matrix S – replace zero rows in H with u=[0. 9, …, 0. 9] – From dangling node, surfer jumps to any other node. 09. 09. 09 I H F L M G B C E D A 0 0 1 0 0 0 0 0 . 5 0 0 0 0 0 . 3 0 0 0 . 5 0 0 1 . 5 0 0 0 0 0 0 0 1 0 0 0 0

Still problems • Random Surfer gets trapped in buckets – Reachable strongly connected component without outlinks • Solution: teleportation matrix E – Matrix of u. 09. 09 I . 09. 09 H F L B C . 09. 09 E . 09. 09. 09 M G D A . 09. 09. 09

… finally • Google matrix G = a. S + (1 -a) E • Where a is the damping factor • Interpretation of G – With probability a, Random Surfer follows a hyperlink from a page (selected at random) – With probability 1 -a, Random Surfer jumps to any page (e. g. , by entering a new URL in the browser) • Page. Rank scores are the solution of self-consistent equation p =p. G =ap. S + (1 -a)u

Page. Rank scores I 1. 6 H 1. 6 L 1. 6 M 1. 6 G 1. 6 E 8. 1 F 3. 9 C 34. 3 B 38. 4 D 3. 9 A 3. 3

Summary • Recursive (or self-consistent) nature of Page. Rank has roots in social network analysis metrics • Page. Rank is fundamentally related to random walks on graphs – Lots of research to compute it efficiently – Huge economic and social impact!